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Let $\pi: E \to X$ be a vector bundle over some manifold $X=\mathcal M$.

Definition 1:

Hatcher defines Chern classes $$c_i(E)\in H^{2i}_\text{singular}(X)$$ as elements of the singular cohomology ring of $X$.

Definition 2:

An invariant polynomial over $\mathfrak{g l}_{m}(\mathbb{C})$ is a polynomial $P$ such that $P(XY)=P(YX)$ for all $X, \, Y \in \mathfrak{g l}_{m}(\mathbb{C})$.

Now, Roe considers generators $P_j$ of the ring of invariant polynomials over $\mathfrak{g l}_{m}(\mathbb{C})$ and $$P_{j}(X)=(-2 \pi i)^{-j} \operatorname{tr}\left(\Lambda^{j} X\right),$$ where $\Lambda^{j} X$ denotes the transformation induced by $X$ on $\Lambda^{j} \mathbb{C}^{m}$.

Suppose $X=\mathcal M$ now is a manifold with a connection $\nabla$ and curvature $K$. Here, $K$ is then a 2-form taking values in $\operatorname{End}(E)$.

Roe defines Chern classes $$c_j(E) := [P_j(K)] \in H^{\mathbb C}_\text{dR}(X),$$ as a de Rahm equivalence class with complex coefficients.

Question:

Are these constructions related, and if so in what way? Considering they lie in different spaces, can we find an isomorphism between them?

What is the link between the Chern-Weil approach to characteristic classes and that of Hatcher?

Markus Klyver
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  • There is a natural map $H^{2i}(X; \mathbb{Z}) \to H^{2i}{\text{dR}}(X)\otimes{\mathbb{R}}\mathbb{C}$. The two definitions of Chern classes correspond to one another under this map. – Michael Albanese Aug 27 '21 at 15:29
  • Hello @MichaelAlbanese, could you please elaborate on this? I of course know about de Rahm's theorem, but it only gives us an isomorphism in the real case and with integer coefficients $H_\text{singular}(X; \mathbb Z) \cong H_\text{dR}^{\mathbb R}(X)$. – Markus Klyver Aug 27 '21 at 19:27
  • What you wrote is incorrect. You get an isomorphism $H^k(X; \mathbb{R}) \to H^k_{\text{dR}}(X)$ from de Rham's Theorem. Now $H^k(X; \mathbb{C}) \cong H^k(X;\mathbb{R})\otimes_{\mathbb{R}}\mathbb{C}\cong H^k_{\text{dR}}(X)\otimes_{\mathbb{R}}\mathbb{C}$. The map $H^k(X; \mathbb{Z}) \to H^k_{\text{dR}}(X)\otimes_{\mathbb{R}}\mathbb{C}$ is the map $H^k(X; \mathbb{Z}) \to H^k(X; \mathbb{C})$ induced by the inclusion $\mathbb{Z} \to \mathbb{C}$ composed with the isomorphism above. – Michael Albanese Aug 27 '21 at 19:36
  • @MichaelAlbanese Could you justify the first isomorphism? – Markus Klyver Aug 28 '21 at 00:24
  • See this answer. – Michael Albanese Aug 28 '21 at 01:05

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